In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly...

in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that...

Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving...

the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense...

R))\to ((P\to Q)\to (P\to R))} ** for deduction theorem (note: {t,b}→{f} = {f} follows from the deduction theorem) ¬ ( P → Q ) → P {\displaystyle \lnot...

intuitionistic logic or intermediate logics and cannot be deduced from the deduction theorem alone. Under the Curry–Howard isomorphism, Peirce's law is the type...

in an ill-formed syllogism, in order to make the form valid. see Deduction theorem Johnson-Laird, Phil (30 December 2009). "Deductive reasoning". WIREs...

implies a negative answer to the Entscheidungsproblem. Using the deduction theorem, the Entscheidungsproblem encompasses the more general problem of...

identities in Heyting algebras. In practice, one frequently uses the deduction theorem in such proofs. Since for any a and b in a Heyting algebra H we have...

for the consistency result, the cut elimination theorem—the Hauptsatz—directly for natural deduction. For this reason he introduced his alternative system...

language with deduction rules. An element ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T {\displaystyle T} is then called a theorem of the...

that the same basic thought (e.g. deduction theorem) must be proven as a metatheorem in Hilbert-style deduction system, while it can be declared explicitly...

Proof: In this proof, universal generalization was used in step 8. The deduction theorem was applicable in steps 10 and 11 because the formulas being moved...

can be restated as shown in the following table. Especially, the deduction theorem specific to Hilbert-style logic matches the process of abstraction...

associated normalization theorem establishes that every derivation in natural deduction can be transformed into normal form. Natural deduction is a system of formal...

theorem (proof theory) Deduction theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem...

way to decompose the resources used by components of a system. The deduction theorem of classical logic relates conjunction and implication: A ∧ B ⊢ C...

tautology (or theorem). Gentzen style. Every line is a conditional tautology (or theorem) with zero or more conditions on the left. Natural deduction. Every...

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories...

these axioms, it is possible to form conservative extensions of the deduction theorem that permit the use of additional connectives. These extensions are...

mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses...

completeness theorem is outlined below. First, using the compactness theorem and the deduction theorem, we may reduce the completeness theorem to its special...

Sebastian (2021). "The Lean 4 Theorem Prover and Programming Language". In Platzer, André; Sutcliffe, Geoff (eds.). Automated Deduction – CADE 28. Lecture Notes...

category theory and related mathematics Deduction Theorem McLarty, Colin (1992). "§17.3 The fundamental theorem". Elementary Categories, Elementary Toposes...

to prove A → C (if A, then C) from the first two premises below: Deduction theorem Logical consequence Propositional calculus Robert L. Causey, Logic...

possible to effectively verify that a purportedly valid deduction is actually a deduction; such deduction systems are called effective. A key property of deductive...

Tolerant sequence Cotolerant sequence Deduction theorem Cirquent calculus Nonconstructive proof Existence theorem Intuitionistic logic Intuitionistic type...

be proved.[citation needed] Examples of metatheorems include: The deduction theorem for first-order logic says that a sentence of the form φ→ψ is provable...

In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf...

JSTOR 2269159. S2CID 250349611. Ruth C. Barcan (December 1946). "The Deduction Theorem in a Functional Calculus of First Order Based on Strict Implication"...